In mathematics, a group action is transitive if it has just one orbit. It is called doubly transitive if it is transitive on ordered pairs of distinct elements; and so on for triply transitive, etc.. An ergodic group action is also called metrically transitive.
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The distinguishing characteristic of a phase transition is an abrupt sudden change in one or more physical properties, in particular the heat capacity, with a small change in a thermodynamic variable such as the temperature.
The various solid/liquid/gas transitions are classified as first-order transitions because they involve a discontinuous change in density (which is the first derivative of the free energy with respect to chemical potential.) Second-order phase transitions have a discontinuity in a second derivative of the free energy.
Universality is a prediction of the renormalization group theory of phase transitions, which states that the thermodynamic properties of a system near a phase transition depend only on a small number of features, such as dimensionality and symmetry, and is insensitive to the underlying microscopic properties of the system.
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